Music Theory Basics

This page collects some notes about scales, intervals etc relevant to the playing of the hurdy-gurdy. It is a work in progress.

Notes

In western music, as taught in the UK we are familiar with a note naming system which includes the notes A, B, C, D, E, F and G.  After  G in the sequence comes another A, described as an octave above the lower A, as it is the eighth note in the sequence.   Each pair of notes is separated by a gap in pitch (denoted an interval) and if we examine the sequence we find that the gaps between consecutive notes are not equal, the gap between A and B is twice that between B and C.  If we call the first of these intervals a tone, and the second a semi-tone, the sequence of notes are separated as follows

A ⇦tone⇨ B ⇦semitone⇨ C ⇦tone⇨ ⇦tone⇨ E ⇦semitone⇨ F ⇦tone⇨ G ⇦tone⇨ A

Each note can be raised by a semitone, the result denoted sharp, or lowered (flat).  From the scale below it is clear that in F flat would be the same as E, and B sharp would be the same note as C. There are 11 distinct pitches, known by their most common names as A, Bb, B, C, C#, D, Eb, F, F#, G and G#. On a modern piano, the sharps and flats are black keys and the remainder are white.

What we have described above is quite specific to modern western music. Dividing the octave into 12 equal pieces, and defining 11 distinct notes is a recent invention, other cultures and periods have had different (and usually more) distinct notes and a greater variety of intervals of between them.  For now it is sensible to accept the above as a labelling of the notes and think how to build scales

Scales and Modes

[At the time of writing this draft note I am not clear of any widely understood and agreed definitions which distinguish scales and modes so will use the term scale. There may be a distinction later].

So far we have a set of notes (11 distinct ones, if we count notes an octave apart as the same note) and they form a sequence of increasing pitch.  Music we would think of as melodic or tuneful rarely uses all of these mixed together. Usually a subset is chosen, for example we can exclude the sharps and flats to get the sequence of the white notes of the piano.  Once we have such a sequence, they can be arranged into a melody and this process introduces another important concept. Usually a melody will have a home note, or root note, to which the tune seems to resolve. Very often a tune will actually end on this note, and may well start on it too. This note is known as the tonic.

When people talk of scales or modes, they are combining these two concepts. If you start by defining the tonic, the sequence is characterised by the intervals you move to create each successive note.

The combination of a particular note sequence, with a melody built around a specific tonic provides a framework for describing a tune. People usually talk of the tune being in a particular key, or mode. The name of that key or mode is made up of two elements, the name of the tonic (for example C) and a word describing the sequence of intervals which gives the rest of the scale or mode (for example “major”).

Clearly this a great simplification, it is worth mentioning just two aspects. The first is that best way to describe part of a tune may not apply to the whole tune, and the term modulation described such a shift. The second is that sometimes the choice of notes is subject to change depending on the context. For example, sometimes an ascending passage will use one note, and a descending passage a different one (often a semi-tone different), and other refinements are possible. But especially for folk melodies, it is a good basis for further discussion.

Keys and Key Signature

At this point we are moving to notation of music. While we can denote any of the 11 tones by a note on a stave, with sharp or flat sign as needed, it is conventional to consider which set of notes the tune uses and to mark the start of each stave with the sharp and flat signs needed to define the sequence, this is called the key signature. For example the sequence of notes (C major scale) C, D, E, F, G, A, B requires an empty key signature (no sharps of flats), where as the same sequence of intervals starting on D requires two (D, E, F#, G, A, B, C#).  It is not conventional to denote the tonic, since it is not needed to play the tune, but it is useful to know what it is, for many reasons. This is particularly true for drone instruments as the most common choice of drone note is indeed the tonic of the melody. This leads us to consider if we can guess the likely tonic of a new piece of music by looking at the key signature (or equivalently, which notes it uses).  In practice, for western folk music we will often be able to narrow down the choice to a small number of possibilities because the same scales or modes occur again and again. Consider the white notes of the piano. In principle a scale could start on any of the 7 notes, and melodies can be built on each of these scales (in this way we get the 7 modes usually denoted with greek names). In western folk however, most of these possibilities are not used, and the likely tonic notes for a tune with no sharps of flats are C  (the major scale, also known as Ionian mode), A (the minor scale, also known as the Aeolian mode), and G (the mixolydian mode). Each of these lends a particular, distinct character to the tune and this combined with the way the tune resolves (often the end note), will usually identify the tonic.

Transposition

Transposition is the process of shifting the  pitch of a tune up or down, while keeping the intervals (or relationship between the notes) the same. Often nowadays tunes are stored in music software and you can just request that the software does the transpose up or down by X semitones. This will shift the notes and change the key signature to match the new key. If you are doing it by hand, it is usually a two step process, transpose the key signature first, then write the notes out the required number of steps above of below where they are in the starting manuscript.

The following chart should help. Find the starting key signature and move up or down by the required number of semitone (each line corresponds to a semitone step).

[table “” not found /]

If you need to extend the table because you run out of rows ou can add or subtract the octave (12 semitones). For example if you want to shift seven semitones down from C major, this would correspond to the row -7 which is not listed. However you can add the octave (12 semitones), -7 + 12 is +5, so the new key is F.

As an example, consider this fragment of a tune (Theme Vannetaise aka Twiglet). It is in the usual key, well suited for D/G melodeons. The tune starts and ends on E, and as it has just one sharp, as such it can be identified as E minor, rows +7 and -5 in the table above.

Let’s suppose we want to play it so that it starts on G (that’s three semitones up, E, -> F -> F# -> G). If we take the -5 row (G major or E minor) in the table and move up three rows we come to the -2 row indicating  that the new key signature has two flats (Bb and Eb), the key signature for Bb major and Gminor.

If every note is moved up three semitones this is the result, using the new key signature

As you can see, in a simple case like this each note has moved up by one line or one space on the stave, which makes it easy to copy the tune at the new pitch if you are doing it by hand. You don’t have to work out the three semitone shift for every note, because the key signature helps you.

 

Fitting tunes onto the hurdy-gurdy

Armed with the information above, we can start thinking about how a tune might be played on the hurdy-gurdy. Most tunes are playable, but for a given tune it may help significantly to transpose the tune first. The factors that affect whether to transpose, and by how much, are as follows

  1. Does the instrument have the required range of notes? It is better to avoid playing a lot at the top or “dusty end” of the instrument if you are a beginner as it is harder to keep these notes sounding good and in tune, and the smaller keys require more accurate finger placement.
  2. Do the drones (including trompette) available harmonise nicely with the tune. Some instruments have capos fitted which allow an easy change of pitch for drones and/or trompette. For those that don’t, changes to the pitch of strings take a while to settle down and also change string tension (which affects the balance of volume of the different strings). Usually drones at at the tonic, or the 5th of the scale work well, drones at the 4th of the scale are possible but the sound is usually less pleasing. D drones therefore work well for tunes in D, G and possibly A (in each case, major or minor).  The C drone will harmonise tunes in C, F and (to an extent) G. Some other combinations work and it is always worth a bit of trial and error.
  3. Is the fingering reasonably comfortable, bearing in mind the ergonomics of the keyboard. To start with, usually it is easiest to use mostly the main row of keys (the C major scale on a G/C instrument, the G major scale on a D/G instrument) although incorporation of a few of the second row keys is not difficult and can even help some fingering sequences. Moving back and forth a lot between the rows a lot is tricky.

D/G Instruments

If you have a D/G gurdy without capos you will normally have drones tuned to D.

The obvious keys given the drone tuning are D major and D minor but for some melodies there is an issue with range. The instrument gives you two octaves of notes starting and ending with D and a D half way up. If a tune has no notes below the tonic it will fit well, you can play in the lower octave and benefit from the harmony of the tonic drone.

The following Bourrée is a good example. Is is written here in D major, the fact that it avoids any notes below the low D reflects the fact that it comes from central French tradition in which the hurdy gurdy is a key instrument.

If the tune has a few notes below the tonic you have to modify the tune, or play it largely in the higher octave, which is not so ergonomic and requires careful instrument setup.

A good example is the Theme Vannetaise used above as a transposition example. If you transpose it down two semitone steps to D minor (the best minor key for D drones), you will have middle C, the first note in the 4th bar which the hurdy-gurdy does not have.

The key of G (major and Minor) is generally very convenient. The tonic of the scale is on the 3rd key on the main row, so you have a few notes below the tonic (D (open string), E, and F# (or F) below. Above the high G, you have a few notes before the keys become very small and awkward.

We transposed Theme Vannetaise into G minor as an example of transposition, above, and you can see there this will easily fit onto a D hurdy-gurdy.

A large fraction of folk tunes fit into this range and with the drone is at the 5th of the scale they are well suited to the D/G hurdy-gurdy.  It is common in the French tradition to find tunes which use all the notes in the scale from G down to D (the fifth below), here is a well-known example.

Tunes in A work OK with the D drone, although the harmony from the drone at the 4th of the A major scale is not as pleasing. As with G, there is a good choice of notes above and below the main octave. A minor is easier than A major from the point of view of fingering.

For the tunes in E minor (such as Theme Vannetaise) it is worth trying to play this in E minor with a D drone. The notes of E minor are the same as G major, and G major tunes sound good with a D drone as discussed above. You can reduce the number of D drones you use (for example play with trompette in D only) but you may find you can play along with melodeons without needing to adjust your instrument. Be guided by your ears (and those of the people playing with you)!

G/C Instruments

G/C instruments have melody strings with G as their open note, and C at the 3rd key. Usually the drones and trompette are in C, often G drones are also fitted and sometimes a G trompette is fitted instead (or as well as) the C one.  The C drones can be retuned (more easily if a capo if fitted) up one note to D

C (major and minor) are probably the optimum keys. You have the benefit of the tonic drone, and also an easily fingered octave from with some notes below and above.

Some tunes in G major and minor work well too, with the proviso mentioned in the case of D tunes on a D/G gurdy – notes below the tonic will push you into the upper octave of the instrument. However many will fit. When playing in G, use G drones if fitted and tune C drones up to D. The drone in D (the 5th in the scale of G) sounds better than C (the 4th).  You will often find that at the higher pitch the drones and trompettes may work a bit better at D due to the higher tension (this depends on the choice of strings fitted to your instrument).

If in doubt, try it out.   You will find the question of whether to transpose a tune is quite subjective. Many people feel that changing the key of the music changes the character of the music in a significant way, even if the tonal shift is quite small.   The sound of a given instrument varies with the chosen note for many reasons, including the fingering and the natural resonances of the instrument,

Beyond Equal temperament

So far we have considered a definition of the pitch of notes in which the octave is divided into 12 equal semitones. This characteristic of most modern instruments (called equal temperament) is in fact a relatively modern practice, it was widely adopted in France and Germany by the late 18th century and in England by the 19th.

It is however a compromise and there are other ways to define the pitches of the notes which generate more harmonious relationships between the pitches. As the hurdy-gurdy is a drone based instrument the important relationship is between each note of the scale and the continuous drone, effectively each note is heard as a two node chord.  The ear tends to hear a harmonious interval when the relationship between two mathematically defined pitch values (measure by the frequency, or the number of vibrations per second) is a simple mathematical ratio. If one notes has twice the frequency of the other, the two notes are so closely related we identify them as different versions of the same note, and we give them the same name… this interval is the octave. Mathematically the ration of the upper frequency to the lower is 2:1.

Other simple ratios also have harmonious sounds, for example a ratio of 3:2 defines the interval of a perfect fifth (7 semitones, from D up to A). Unlike the octave however, the frequencies of a modern piano or accordion for the notes A and D are not in the exact ratio of 3:2, as this doesn’t fit the division of the octave into 12 equal semitones.  Although the difference is small it does give us a choice when tuning the hurdy-gurdy, if we adjust the tangent for the A key to get the best harmony with the D drone our ears will guide us towards the more exact 3:2 ratio, whereas using an electronic tuner will give us the version of A consistent with modern, equal tempered instruments.

As we move through the scale, we find that in each case there is a way to define the note using the relationship to the tonic drone note (defined by a ration x:y where x and y are small numbers) and there is a way to define it by adding the appropriate number of equal sized semitones. For some notes on the scale the differences between these two approaches are significant.  The definition of the notes using simple ratios is called just temperament and these intervals really do sound more harmonious.

It is possible to tune the notes of a hurdy-gurdy using to just temperament using an electronic tuner, provided the tuner allows you to read off how far above or below the equal tempered pitch value the sound is.  A table to allow you to do this is provided below. To understand the adjustments, you need to understand that the tuner will report the difference in pitch as a number of cents. A cent is one hundredth of a semitone, so there are 1200 cents to an octave.

Just Intonation

SEMITONE D SCALEG SCALECENTSJUST RATIO
0DG01:1
1EbAb+1216:15
2EA+49:8
3FBb+166:5
4F#B-145:4
5GC-24:3
6G#C#-1045:32
7AD+23:2
8BbEb+148:5
9BE-165:3
10CF-416:9
11C#F#-1215:8
12DG02:1

If you tune your instrument in this way you will find it sounds significantly better than using equal temperament when playing alone (or accompanying a song). When playing with other equal tempered instruments there may be some notes which clash, depending on the type of music and how it is arranged you may want to compromise and retune some notes to match your fellow musicians.

The main drawback of just temperament, and the reason it is no longer popular is that while it enhances the sound when playing in the home key (usually the key of the drones) this benefit is only fully realised in that one tonality. If you shift to another key, especially one that is not closely related (by an interval of a 4th or 5th) the benefits are lost and the some intervals will sound worse than equal temperament. Equal temperament is a compromise to allow an instrument to play equally well (or equally badly, depending on your point of view!) in all keys.

Links to Other Resources

Neil Brook’s hurdy-gurdy maintenance tips  http://www.hurdy-gurdy.org.uk/hurdy%20gurdy%20tips.html

More detailed discussions on Tuning and Temperament

http://www.orfeo-fiato.nl/hurdy_gurdy_tuning.html

http://www.hurdygurdy.org/pdfs/tuningandtemperament.pdf

http://www.phy.mtu.edu/~suits/scales.html

http://www.yuvalnov.org/temperament/